Nooooooo. That's so...weird. Maybe, if you learned something about the field of medicine in their care there, it could be okay to mention your experience with the healthcare system. But specifying the institution you are applying to is...a weird form of name dropping. And they're not gonna go verify that information, they'll probably just think it was a weird thing to say.
It wouldn't be too weird for something like:
Interviewer: "Anything else you would like to mention?"
You:"As my little brother has been undergoing months of treatment here for his Hosgkinsretinosarcoblastoma, I have seen the care and compassion of your medical students, residents, and attendings. Their excellence further inspires me. I have also had to be support for him and my mother through this, and hope to continue that in the future. In addition, I have seen first hand the effects of your cutting edge research in NanoMolecule27, which is related to MicroParticle32, about which I have done extensive research in undergrad...Therefore, I think I am an excellent fit for this institution."
It would be pretty weird for something like:
Interviewer: "Do you have any ties to the area?"
You:"My second cousin is being treated for herpes here."
You: "Oh, and once my uncle had a rectal abscess drained here."
Interviewer *Blink* *Blink*
You: "So, I'd say I have some pretty significant ties here."
It's all in how you sell it. But I think unless it demonstrates you have a significant need to be in the area or allows you speak to specific things you know about the institution, it probably doesn't matter.